Elliptic Curves and Algebraic Geometry Approach in Gravity Theory I.The General Approach

TL;DR

This paper explores the algebraic structure of gravity theories using affine geometry, revealing the existence of high-degree algebraic equations and proposing new solution methods for Einstein's equations.

Contribution

It introduces a general algebraic approach to gravity theory based on covariant and contravariant metrics, including new algebraic equations and properties of a generalized connection.

Findings

Existence of high-degree algebraic equations in gravity theory.

Derivation of algebraic equations for contravariant tensor components.

The generalized connection has affine transformation and equiaffine properties.

Abstract

Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third, fourth, fifth, sixth, seventh - degree algebraic equations exists in gravity theory. This fact, together with the derivation of the algebraic equations for a generally defined contravariant tensor components in this paper, are important in view of finding new solutions of the Einstein's equations, if they are treated as algebraic ones. Some important properties of the introduced in hep-th/0107231 more general connection have been also proved - it possesses affine transformation properties and it is an equiaffine one. Basic and important knowledge about the affine geometry approach and about gravitational theories with covariant and contravariant connections and metrics is also given with the purpose of demonstrating when and how these theories can be related to the proposed algebraic approach and to the existing theory of gravity and relativistic hydrodynamics.